#8184924

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,6)(4,8)(5,13)(7,10)(9,15)(11,16)(12,14)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34)(35,45)(36,48)(37,43)(38,42)(39,44)(40,47)(41,46);(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,16)(13,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35)(34,40)(36,37)(38,46)(39,47)(41,43)(42,48)(44,45);(1,3,7,4)(2,8,10,6)(5,11,15,12)(9,16,13,14)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,36,39,46)(34,41,44,48)(35,37,47,38)(40,43,45,42);(1,4,7,3)(2,8,10,6)(5,12,15,11)(9,16,13,14)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,37,39,38)(34,42,44,43)(35,36,47,46)(40,48,45,41);(1,5,7,15)(2,9,10,13)(3,11,4,12)(6,14,8,16)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32)(33,38,39,37)(34,43,44,42)(35,46,47,36)(40,41,45,48);(1,6,7,8)(2,3,10,4)(5,14,15,16)(9,11,13,12)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,34)(35,40)(36,48)(37,42)(38,43)(39,44)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092120);